Let $\mathcal{D}=\{D_0,\ldots,D_{n-1}\}$ be a set of $n$ topological disks in the plane and let $\mathcal{A} := \mathcal{A}(\mathcal{D})$ be the arrangement induced by~$\mathcal{D}$. For two disks $D_i,D_j\in\mathcal{D}$, let $\Delta_{ij}$ be the number of connected components of~$D_i\cap D_j$, and let $\Delta := \max_{i,j} \Delta_{ij}$. We show that the diameter of $\mathcal{G}^*$, the dual graph of~$\mathcal{A}$, can be bounded as a function of $n$ and $\Delta$. Thus, any two points in the plane can be connected by a Jordan curve that crosses the disk boundaries a number of times bounded by a function of~$n$ and~$\Delta$. In particular, for the case of two disks we prove that the diameter of $\mathcal{G}^*$ is at most $\max\{2,2\Delta\}$ and this bound is tight. % For the general case of $n>2$ disks, we show that the diameter of $\mathcal{G}^*$ is at most $2 n(\Delta+1)^{n(n-1)/2} \min\{n,\Delta+1\}$. We achieve this by proving that the number of maximal faces in $\mathcal{A}$ -- the faces whose ply is more than the ply of their neighboring faces -- is at most $n(\Delta+1)^{n(n-1)/2}$, which is interesting in its own right.

翻译：设 $\mathcal{D}=\{D_0,\ldots,D_{n-1}\}$ 为平面上 $n$ 个拓扑圆盘的集合，并令 $\mathcal{A} := \mathcal{A}(\mathcal{D})$ 为由 $\mathcal{D}$ 诱导的构型。对于任意两个圆盘 $D_i,D_j\in\mathcal{D}$，令 $\Delta_{ij}$ 表示 $D_i\cap D_j$ 的连通分支数，并令 $\Delta := \max_{i,j} \Delta_{ij}$。我们证明了 $\mathcal{A}$ 的对偶图 $\mathcal{G}^*$ 的直径可以被 $n$ 和 $\Delta$ 的函数所界定。因此，平面上任意两点可由一条若尔当曲线连接，且该曲线穿越圆盘边界的次数被 $n$ 和 $\Delta$ 的某个函数所限定。特别地，对于两个圆盘的情形，我们证明了 $\mathcal{G}^*$ 的直径至多为 $\max\{2,2\Delta\}$，且该界是紧的。对于 $n>2$ 个圆盘的一般情形，我们证明了 $\mathcal{G}^*$ 的直径至多为 $2 n(\Delta+1)^{n(n-1)/2} \min\{n,\Delta+1\}$。我们通过证明 $\mathcal{A}$ 中最大面（即其层数大于相邻面层数的面）的数量至多为 $n(\Delta+1)^{n(n-1)/2}$ 来达成上述结论，这一结果本身亦具有独立意义。